In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell. It is named so because it is a strengthening of Minkowski's theorem.
Setting Let K be a closed convex centrally symmetric body of positive finite volume in n-dimensional Euclidean space Rn. The gauge or distance Minkowski functional g attached to K is defined by
g ( x ) = inf { λ ∈ R : x ∈ λ K } . {\displaystyle g(x)=\inf \left\{\lambda \in \mathbb {R} :x\in \lambda K\right\}.}
Conversely, given a norm g on Rn we define K to be
K = { x ∈ R n : g ( x ) ≤ 1 } . {\displaystyle K=\left\{x\in \mathbb {R} ^{n}:g(x)\leq 1\right\}.}
Let Γ be a lattice in Rn. The successive minima of K or g on Γ are defined by setting the k-th successive minimum λk to be the infimum of the numbers λ such that λK contains k linearly-independent vectors of Γ. We have 0 < λ1 ≤ λ2 ≤ ... ≤ λn < ∞.
Statement The successive minima satisfy
2 n n ! vol ( R n / Γ ) ≤ λ 1 λ 2 ⋯ λ n vol ( K ) ≤ 2 n vol ( R n / Γ ) . {\displaystyle {\frac {2^{n}}{n!}}\operatorname {vol} \left(\mathbb {R} ^{n}/\Gamma \right)\leq \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} \left(\mathbb {R} ^{n}/\Gamma \right).}
Proof A basis of linearly independent lattice vectors b1, b2, ..., bn can be defined by g(bj) = λj (warning : it may be not a basis of the lattice, but only a basis of the ambient space). The lower bound is proved by considering the convex polytope 2n with vertices at ±bj/ λj, which has an interior enclosed by K and a volume which is 2n/n!λ1 λ2...λn times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λj along each basis vector to obtain 2n n-simplices with lattice point vectors), the integer being the index of the Z {\displaystyle \mathbb {Z} } -span of the family ( b i ) i {\displaystyle (b_{i})_{i}} as a subgroup of the lattice. To prove the upper bound, consider functions fj(x) sending points x in K {\textstyle K} to the centroid of the subset of points in K {\textstyle K} that can be written as x + ∑ i = 1 j − 1 a i b i {\textstyle x+\sum _{i=1}^{j-1}a_{i}b_{i}} for some real numbers a i {\textstyle a_{i}} . Then the coordinate transform x ′ = h ( x ) = ∑ i = 1 n ( λ i − λ i − 1 ) f i ( x ) / 2 {\displaystyle x'=h(x)=\sum _{i=1}^{n}(\lambda _{i}-\lambda _{i-1})f_{i}(x)/2} has a Jacobian determinant J = λ 1 λ 2 … λ n / 2 n {\textstyle J=\lambda _{1}\lambda _{2}\ldots \lambda _{n}/2^{n}} . If p {\textstyle p} and q {\textstyle q} are in the interior of K {\textstyle K} and p − q = ∑ i = 1 k a i b i {\textstyle p-q=\sum _{i=1}^{k}a_{i}b_{i}} (with a k ≠ 0 {\textstyle a_{k}\neq 0} ) then ( h ( p ) − h ( q ) ) = ∑ i = 0 k c i b i ∈ λ k K {\displaystyle (h(p)-h(q))=\sum _{i=0}^{k}c_{i}b_{i}\in \lambda _{k}K} with c k = λ k a k / 2 {\textstyle c_{k}=\lambda _{k}a_{k}/2} , where the inclusion in λ k K {\textstyle \lambda _{k}K} (specifically the interior of λ k K {\textstyle \lambda _{k}K} ) is due to convexity and symmetry. But lattice points in the interior of λ k K {\textstyle \lambda _{k}K} are, by definition of λ k {\textstyle \lambda _{k}} , always expressible as a linear combination of b 1 , b 2 , … b k − 1 {\textstyle b_{1},b_{2},\ldots b_{k-1}} , so any two distinct points of K ′ = h ( K ) = { x ′ ∣ h ( x ) = x ′ } {\textstyle K'=h(K)=\{x'\mid h(x)=x'\}} cannot be separated by a lattice vector. Therefore, K ′ {\textstyle K'} must be enclosed in a primitive cell of the lattice (which has volume vol ( R n / Γ ) {\textstyle \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )} ), and consequently vol ( K ) / J = vol ( K ′ ) ≤ vol ( R n / Γ ) {\textstyle \operatorname {vol} (K)/J=\operatorname {vol} (K')\leq \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )} .
References
Cassels, J. W. S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. Zbl 0077.04801. Cassels, J. W. S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN 978-3-540-61788-4. Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. pp. 180–185. ISBN 0-387-94655-1. Zbl 0859.11003. Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 6. ISBN 3-540-54058-X. Zbl 0754.11020. Siegel, Carl Ludwig (1989). Komaravolu S. Chandrasekharan (ed.). Lectures on the Geometry of Numbers. Springer-Verlag. ISBN 3-540-50629-2. Zbl 0691.10021.